Multiple regression principles (Notes)

STAT 155

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Notes

Learning goals

Working with multiple predictors in our plots and models can get complicated!

There are no recipes for this process.

BUT there are some guiding principles that assist in long-term retention, deeper understanding, and the ability to generalize our tools in new settings.

By the end of this lesson, you should be familiar with some general principles for…

  • incorporating additional quantitative or categorical predictors in a visualization
  • how additional quantitative or categorical predictors impact the physical representation of a model
  • interpreting quantitative or categorical coefficients in a multiple regression model

Readings and videos

Please watch the following video before class.

Exercises

Let’s revisit the bikeshare data:

# Load packages & import data
library(readr)
library(ggplot2)
library(dplyr)

bikes <- read_csv("https://mac-stat.github.io/data/bikeshare.csv") %>% 
  rename(rides = riders_registered)

Our goal is to understand how / why registered ridership from day to day.

To this end, we’ll build various multiple linear regression models of rides by different combinations of the possible predictors.

# Check out the data
head(bikes)
## # A tibble: 6 × 15
##   date       season  year month day_of_week weekend holiday temp_actual
##   <date>     <chr>  <dbl> <chr> <chr>       <lgl>   <chr>         <dbl>
## 1 2011-01-01 winter  2011 Jan   Sat         TRUE    no             57.4
## 2 2011-01-02 winter  2011 Jan   Sun         TRUE    no             58.8
## 3 2011-01-03 winter  2011 Jan   Mon         FALSE   no             46.5
## 4 2011-01-04 winter  2011 Jan   Tue         FALSE   no             46.8
## 5 2011-01-05 winter  2011 Jan   Wed         FALSE   no             48.7
## 6 2011-01-06 winter  2011 Jan   Thu         FALSE   no             47.1
## # ℹ 7 more variables: temp_feel <dbl>, humidity <dbl>, windspeed <dbl>,
## #   weather_cat <chr>, riders_casual <dbl>, rides <dbl>, riders_total <dbl>

Exercise 1: Review visualization

Let’s build a model of rides by windspeed (quantitative) and weekend status (categorical).

  1. Write a model statement for this regression model.

  2. Plot & describe, in words, the relationship between these 3 variables.

# Plot of rides vs windspeed & weekend
# HINT: Start with a plot of rides vs windspeed, then add an aesthetic for weekend!

Exercise 2: Review model

Let’s build the model. Run the following code:

bike_model_1 <- lm(rides ~ windspeed + weekend, data = bikes)
coef(summary(bike_model_1))
##               Estimate Std. Error   t value      Pr(>|t|)
## (Intercept) 4738.38053  147.53653 32.116659 1.208405e-141
## windspeed    -63.97072   10.45274 -6.119997  1.528443e-09
## weekendTRUE -925.15701  119.86330 -7.718434  3.891082e-14

The model formula with our coefficient estimates filled in is therefore:

E[rides | windspeed, weekendTRUE] = 4738.38 - 63.97 * windspeed - 925.16 * weekendTRUE

This model formula is represented by 2 lines, one corresponding to weekends and the other to weekdays. Simplify the model formula above for weekdays and weekends:

weekdays: rides = ___ - ___ windspeed

weekends: rides = ___ - ___ windspeed

Exercise 3: Review coefficient interpretation

  1. The intercept coefficient, 4738.38, represents the intercept of the sub-model for weekdays, the reference category. What’s its contextual interpretation?

  2. The windspeed coefficient, -63.97, represents the shared slope of the weekend and weekday sub-models. What’s its contextual interpretation?

  3. The weekendTRUE coefficient, -925.16, represents the change in intercept for the weekend vs weekday sub-model. What’s its contextual interpretation?

Exercise 4: 2 categorical predictors – visualization

Thus far, we’ve explored a couple examples of multiple regression models that have 2 predictors, 1 quantitative and 1 categorical.

So what happens when both predictors are categorical?!

To this end, let’s model rides by weekend status and season.

The below code plots rides vs season.

Modify this code to also include information about weekend.

HINT: Remember the visualization principle that additional categorical predictors require some sort of grouping mechanism / mechanism that distinguishes between the 2 groups.

# rides vs season
bikes %>% 
  ggplot(aes(y = rides, x = season)) + 
  geom_boxplot()

# rides vs season AND weekend
bikes %>%
  ggplot(aes(y = rides, x = season, ___ = ___)) +
  geom_boxplot()
## Error in parse(text = input): <text>:8:38: unexpected input
## 7: bikes %>%
## 8:   ggplot(aes(y = rides, x = season, __
##                                         ^

Exercise 5: follow-up

  1. Describe (in words) the relationship of ridership with season & weekend status.

  2. A model of rides by season alone would be represented by only 4 expected outcomes, 1 for each season. Considering this and the plot above, how do you anticipate a model of rides by season and weekend status will be represented?

    • 2 lines, 1 for each weekend status
    • 8 lines, 1 for each possible combination of season & weekend
    • 2 expected outcomes, 1 for each weekend status
    • 8 expected outcomes, 1 for each possible combination of season & weekend

Exercise 6: 2 categorical predictors – build the model

Let’s build the multiple regression model of rides vs season and weekend:

bike_model_2 <- lm(rides ~ weekend + season, bikes)
coef(summary(bike_model_2))
##                Estimate Std. Error     t value      Pr(>|t|)
## (Intercept)   4260.4492   99.16363  42.9638294 1.384994e-201
## weekendTRUE   -912.3324  103.23016  -8.8378473  7.298199e-18
## seasonspring  -116.3824  132.76018  -0.8766364  3.809741e-01
## seasonsummer   438.4424  132.06413   3.3199205  9.454177e-04
## seasonwinter -1719.0572  133.30505 -12.8956646  2.081758e-34

Thus the model formula with coefficient estimates filled in is given by:

E[rides | weekend, season] = 4260.45 - 912.33 weekendTRUE - 116.38 seasonspring + 438.44 seasonsummer - 1719.06 seasonwinter

  1. Use this model to predict the ridership on the following days:
# a fall weekday
4260.45 - 912.33*___ - 116.38*___  + 438.44*___ - 1719.06*___

# a winter weekday    
4260.45 - 912.33*___ - 116.38*___  + 438.44*___ - 1719.06*___

# a fall weekend day        
4260.45 - 912.33*___ - 116.38*___  + 438.44*___ - 1719.06*___

# a winter weekend day
4260.45 - 912.33*___ - 116.38*___  + 438.44*___ - 1719.06*___
## Error in parse(text = input): <text>:2:19: unexpected input
## 1: # a fall weekday
## 2: 4260.45 - 912.33*__
##                      ^
  1. We only made 4 predictions here. How many possible predictions does this model produce? Is this consistent with your intuition in the previous exercise?

Exercise 7: 2 categorical predictors – interpret the model

Use your above predictions and visualization to fill in the below interpretations of the model coefficients.

Hint: What is the consequence of plugging in 0 or 1 for the different weekend and season categories?

  1. Interpreting 4260: On average, we expect there to be 4260 riders on (weekdays/weekends) during the (fall/spring/summer/winter).

  2. Interpreting -912: On average, in any season, we expect there to be 912 (more/fewer) riders on weekends than on ___.

An alternative interpretation: On average, we expect there to be 912 (more/fewer) riders on weekends than on ___, adjusting for season.

  1. Interpreting -1719: On average, on both weekdays and weekends, we expect there to be 1719 (more/fewer) riders in winter than in ___.

An alternative interpretation: On average, we expect there to be 1719 (more/fewer) riders in winter than in ___, controlling for weekday status.

Exercise 8: 2 quantitative predictors – visualization

Next, consider the relationship between rides and 2 quantitative predictors: windspeed and temp_feel. Check out the plot of this relationship below.

This reflect the visualization principle that quantitative variables require some sort of numerical scaling mechanism – rides and windspeed get numerical axes, and temp_feel gets a color scale.

Modify the code below to recreate this plot.

bikes %>%
  ggplot(aes(y = rides, x = windspeed, ___ = ___)) +
  geom_point()
## Error in parse(text = input): <text>:2:41: unexpected input
## 1: bikes %>%
## 2:   ggplot(aes(y = rides, x = windspeed, __
##                                            ^

Exercise 9: follow-up

Describe (in words) the relationship of ridership with windspeed & temperature.

Exercise 10: 2 quantitative predictors – modeling

Let’s build the multiple regression model of rides vs windspeed and temp_feel:

bike_model_3 <- lm(rides ~ windspeed + temp_feel, data = bikes)
coef(summary(bike_model_3))
##              Estimate Std. Error     t value     Pr(>|t|)
## (Intercept) -24.06464 299.303032 -0.08040225 9.359394e-01
## windspeed   -36.54372   9.408116 -3.88427585 1.119805e-04
## temp_feel    55.51648   3.330739 16.66791759 4.436963e-53

Thus the model formula with coefficient estimates filled in is given by,

E[rides | windspeed, temp_feel] = -24.06 - 36.54 windspeed + 55.52 temp_feel

  1. Interpret the intercept coefficient, -24.06, in context.

  2. Interpret the windspeed coefficient, -36.54, in context.

  3. Interpret the temp_feel coefficient, 55.52, in context.

Exercise 11: Which is “best”?

We’ve now observed 3 different models of ridership, each having 2 predictors. The R-squared values of these models, along with those of the simple linear regression models with each predictor alone, are summarized below.

model predictors R-squared
bike_model_1 windspeed & weekend 0.119
bike_model_2 weekend & season 0.349
bike_model_3 windspeed & temp_feel 0.310
bike_model_4 windspeed 0.047
bike_model_5 temp_feel 0.296
bike_model_6 weekend 0.074
bike_model_7 season 0.279

  1. Which model does the best job of explaining the variability in ridership from day to day?

  2. If you could only pick one predictor, which would it be?

  3. What happens to R-squared when we add a second predictor to our model, and why does this make sense? For example, how does the R-squared for model 1 (with both windspeed and weekend) compare to those of model 4 (only windspeed) and model 6 (only weekend)?

  4. Are 2 predictors always better than 1? Provide evidence and explain why this makes sense.

Exercise 12: Principles of interpretation

These exercises have revealed some principles behind interpreting model coefficients, summarized below.

Review and confirm that these make sense.

Principles of interpretation

Consider a multiple linear regression model:

\[E[Y | X_1, X_2, ..., X_p] = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + ... + \beta_p X_p\]

We can interpret the coefficients as follows:

  • \(\beta_0\) (“beta 0”) is the y-intercept. It describes the average value of \(Y\) when \(X_1, X_2,..., X_k\) are all 0, ie. when all quantitative predictors are set to 0 and the categorical predictors are set to their reference levels.

  • \(\beta_i\) (“beta i”) is the coefficient of \(X_i\).

    • If \(X_i\) is quantitative, \(\beta_i\) describes the average change in \(Y\) associated with a 1-unit increase in \(X_i\) while at a fixed set of the other \(X\).

    • If \(X_i\) represents a category of a categorical variable, \(\beta_i\) describes the average difference in \(Y\) between this category and the reference category, while at a fixed set of the other \(X\).





Extra practice

The following exercises provide extra practice. If you don’t get to these during class, you’re encouraged to try them outside class.

Exercise 13: Practice 1

Consider the relationship of rides vs weekend and weather_cat.

  1. Construct a visualization of this relationship.
  2. Construct a model of this relationship.
  3. Interpret the first 3 model coefficients.

Exercise 14: Practice 2

Consider the relationship of rides vs temp_feel and humidity.

  1. Construct a visualization of this relationship.
  2. Construct a model of this relationship.
  3. Interpret the first 3 model coefficients.

Exercise 15: Practice 3

Consider the relationship of rides vs temp_feel and weather_cat.

  1. Construct a visualization of this relationship.
  2. Construct a model of this relationship.
  3. Interpret the first 3 model coefficients.

Exercise 16: CHALLENGE

We’ve explored models with 2 predictors. What about 3 predictors?! Consider the relationship of rides vs temp_feel, humidity, AND weekend.

  1. Construct a visualization of this relationship.
  2. Construct a model of this relationship.
  3. Interpret each model coefficient.

Done!

  • Finalize your notes: (1) Render your notes to an HTML file; (2) Inspect this HTML in your Viewer – check that your work translated correctly; and (3) Outside RStudio, navigate to your ‘Activities’ subfolder within your ‘STAT155’ folder and locate the HTML file – you can open it again in your browser.
  • Clean up your RStudio session: End the rendering process by clicking the ‘Stop’ button in the ‘Background Jobs’ pane.
  • Check the solutions in the course website, at the bottom of the corresponding chapter.
  • Work on homework!